1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A conjecture about Dirichlet series.

  1. Mar 27, 2007 #1

    tpm

    User Avatar

    if [tex] g(s)= \sum_{n=1}^{\infty} a(n) n^{-s} [/tex]

    Where g(s) has a single pole at s=1 with residue C, then my question/conjecture is if for s >0 (real part of s bigger than 0) we can write

    [tex] g(s)= C(\frac{1}{s-1}+1)-s\int_{0}^{\infty}dx(Cx-A(x))x^{-s-1} [/tex]

    of course [tex] A(x)=\sum_{n \le x}a(n) [/tex]

    the question is if the series converge for s >1 with a pole there is a method to 'substract' this singularity (pole) at s=1 to give meaning for the series at any positive s.

    I think that the 'Ramanujan resummation' may help to give the result:

    [tex] \sum_{ n >1}^{[R]}a(n)n^{-s} = g(s)-C(s-1)^{-1} [/tex] valid even for s=1 or s>0 (??)
     
    Last edited: Mar 27, 2007
  2. jcsd
  3. Mar 27, 2007 #2
    Jose, how's it been?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: A conjecture about Dirichlet series.
  1. Dirichlet series (Replies: 1)

  2. Dirichlet series (Replies: 5)

Loading...